Description of Adiabatic Expansion

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Discussion Overview

The discussion centers on the derivation and relationship between the equations for adiabatic expansion of an ideal gas, specifically the expressions ## TV^{\gamma - 1} = constant ## and ## pV^{\gamma} = constant ##. Participants explore how to transition from one equation to the other and the implications of constants involved in the derivation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant understands the derivation of ## TV^{\gamma - 1} = constant ## but seeks clarity on how it leads to ## pV^{\gamma} = constant ##.
  • Another participant suggests substituting ## \frac{PV}{nR} ## for ## T ## in the first equation, proposing that this could yield the second equation if the number of moles ## n ## remains constant.
  • A subsequent reply indicates that this substitution results in ## pV^{\gamma} = R(constant) ##, implying a new constant that incorporates ## R ##.
  • Some participants question whether this derivation is straightforward and suggest that access to a textbook might be necessary for clarity.
  • There is a mention that the new constant in the second equation is different, being the original constant multiplied by ## nR ##.
  • Several participants recommend online resources, such as the Hyperphysics website, for further information on gas laws.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the derivation process, with some suggesting it is straightforward while others indicate confusion. There is no consensus on the ease of the derivation or the necessity of external resources.

Contextual Notes

Participants acknowledge the presence of constants in the equations but do not resolve how to manipulate these constants in the derivation. The discussion reflects uncertainty about the derivation process and the accessibility of resources.

I_laff
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I've seen the derivation for the adiabatic expansion of an ideal gas which gives the result ## TV^{\gamma - 1} = constant ## which I understand. I have also seen the a similar result, ## pV^{\gamma} = constant ##. But I can't see how to get from the first expression to the second. Any ideas?
 
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If you put ##\frac{PV}{nR}## in place of ##T## in the first equation, doesn't it become the second one, assuming that the number of moles ##n## remains constant in the expansion?
 
From doing that you get ## pV^{\gamma} = R(constant) ##. So you just define a new constant on the RHS that contains ## R ##?
 
Isn't this a straightforward bit of Text Book derivation? Do you not have access to one?
 
Yes, it's a different constant then, the original one multiplied by ##nR##.
 
If you don't have a textbook, try the Hyperphysics website. They have a fair amount of stuff on the gas laws.
 
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sophiecentaur said:
Isn't this a straightforward bit of Text Book derivation? Do you not have access to one?
You are probably right, however I don't have a textbook on thermodynamics. I thought of substituting ## \frac{pV}{nR} ## but didn't see how to remove ## nR ## from the final expression. Since they're constant, I guess it's obvious the new constant contains these terms.
 
sophiecentaur said:
If you don't have a textbook, try the Hyperphysics website. They have a fair amount of stuff on the gas laws.
Thanks, I'll check it out :thumbup:.
 
I_laff said:
Thanks, I'll check it out :thumbup:.
There are other on-line sources which are an alternative to a textbook but Hyperphysics is fairly user friendly.
 
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